center of gravity

[crazyvirus85]

A region R consists of a square bounded by the lines x = -8, x = 8, y = 0, and y = -16 and a half disk bounded by the semicircle y = sqrt(64 - x^2) and the line y = 0. Find the center of gravity ( x y ). (x-bar, y-bar)

 

[soroban]

Hello, crazyvirus85!

A region R consists of a square bounded by the lines x = -8, x = 8, y = 0, and y = -16
and a half disk bounded by the semicircle y = sqrt(64 - x²) and the line y = 0.

Find the center of gravity (x-bar, y-bar)

Here's a start on it . . .

Assuming the region has uniform density, the CG is on the y-axis.

The CG of a semicircle is 4r/3π units up from the center.
The CG of your semicircle is (0,32/3π) **

The CG of the square is (0,-8), of course.

Do you know how to "combine" them?

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

** That is <u>not</u> sloppy typing . . . the π (pi) is in the denominator.

 

[crazyvirus85]

so do u just add the CGs or do u also take the integral?